First, you only need 20% diving skill for amphibious assault. Second, to increase your diving skill jump in the water anywhere and repeat 100 times - dive, come up, dive, come up, dive, come up and so on. Every once in a while you'll see a bar showing your diving skill increasing. That's a 5% increase, so you need four of those. Now, and this is important - it doesn't help to dive for a long time. The game doesn't give you extra credit. Instead, it just counts the number of times you've dived. So dive again and again and again until you get there.

If you want to up your diving skill and do something useful, download a map of all the oysters, and go collecting oysters. Take a boat for the trip and dive near each oyster to pick it up. Mark each oyster you get on the map so you don't lose track.

[Correction: it goes up 5% every time you see "lung capacity upgraded", not every time you see the green bar with a plus]